The mean and population variance of a data 3,8,9,x1, and x2 are 9 and 13.2. respectively. Find the 75th percentile for the given dataset.?
a.14
b.11
c.9
d.3
Answers
Given : The mean and population variance of a data 3 , 8 , 9 , x₁ , x₂ are 9 and 13.2
To Find : 75th percentile of given data
14
11
9
3
Solution:
Mean = 9
population variance = 13.2
Mean = ∑ xi / n
=> ∑ xi = n * Mean
3 + 8 + 9 + x₁ + x₂ = 5 * 9
=> x₁ + x₂ = 25
=> x₂ = 25 - x₁
∑(xi - Mean)² = (3 -9)² + (8 - 9)² + (9 - 9)² + (x₁ - 9)² + (x₂-9)²
= 36 + 1 + (x₁ - 9)² + (x₂-9)²
= 37 + (x₁ - 9)² + (25-x₁ -9)²
= 37 + (x₁ - 9)² + (16-x₁ )²
= 37 + x₁² -18x₁ + 81 + 256 -32x₁ + x₁²
= 374 + 2x₁² - 50x₁
population variance = ∑(xi - Mean)² /n
=> 13.2 = ∑(xi - Mean)² /5
=> 66 = ∑(xi - Mean)²
=> 66 = 374 + 2x₁² - 50x₁
=> 2x₁² - 50x₁ + 308 = 0
=> x₁² -25x₁ + 154 = 0
=> (x₁ - 11) (x₁ - 14) = 0
=> x₁ = 11 or 14
Then x₂ = 14 or 11
Hence data set is
3 , 8 , 9 , 11 , 14 in ascending order
Q₁ = 8 (25th percentile )
Q₂ = Median = 9 (50th percentile )
Q₃ = 11 ( 75th percentile)
Hence 75th percentile is 11
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